Properties

Label 714f
Number of curves $4$
Conductor $714$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 714f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
714.e4 714f1 \([1, 1, 1, 1, 101]\) \(103823/4386816\) \(-4386816\) \([4]\) \(192\) \(-0.046597\) \(\Gamma_0(N)\)-optimal
714.e3 714f2 \([1, 1, 1, -319, 2021]\) \(3590714269297/73410624\) \(73410624\) \([2, 2]\) \(384\) \(0.29998\)  
714.e2 714f3 \([1, 1, 1, -679, -3883]\) \(34623662831857/14438442312\) \(14438442312\) \([2]\) \(768\) \(0.64655\)  
714.e1 714f4 \([1, 1, 1, -5079, 137205]\) \(14489843500598257/6246072\) \(6246072\) \([2]\) \(768\) \(0.64655\)  

Rank

sage: E.rank()
 

The elliptic curves in class 714f have rank \(1\).

Complex multiplication

The elliptic curves in class 714f do not have complex multiplication.

Modular form 714.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - q^{12} - 6q^{13} - q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.